Statistical problems with high-dimensional data are frequently plagued by the curse of dimensionality, in which the number of samples required to solve the problem grows rapidly with the dimensionality of the input. The classical theory explains this phenomenon as the consequence of basic geometric and algebraic properties of high-dimensional spaces; for example, the number of \(\epsilon \) -cubes inside a unit cube in \(\mathbb {R}^{d}\)  grows exponentially like \(\epsilon ^{-d}\) , and the number of degree \(r\)  polynomials in \(\mathbb {R}^d\)  grows like a power-law \(d^r\) . Since for real-world problems \(d\)  is typically in the hundreds or thousands, classical wisdom suggests that learning is likely to be infeasible. However, building off of the groundbreaking work AlexNet [1]}, practitioners in deep learning have tackled a wide range of difficult real-world learning problems [2]}, [3]}, [4]}, [5]}, [6]} in high dimensions, once believed by many to be out-of-scope of current techniques. The astonishing success of modern machine learning methods clearly contradicts the curse of dimensionality and therefore poses the fundamental question: mathematically, how do modern machine learning methods break the curse of dimensionality?
